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Potential Difference

Potential Difference

When a circuit is connected and complete, charge can move through the circuit. Charge will not move unless there is a reason, a force. Think of it as though charge is at rest and something has to push it along. This means that work needs to be done to make charge move. A force acts on the charges, doing work, to make them move. The force is provided by the battery in the circuit.

We call the moving charge "current" and we will talk about this later.

The position of the charge in the circuit tells you how much potential energy it has because of the force being exerted on it. This is like the force from gravity, the higher an object is above the ground (position) the more potential energy it has.

The amount of work to move a charge from one point to another point is how much the potential energy has changed. This is the difference in potential energy, called potential difference. Notice that it is a difference between the value of potential energy at two points so we say that potential difference is measured between or across two points. We do not say potential difference through something.

Definition 1: Potential Difference

Electrical potential difference as the difference in electrical potential energy per unit charge between two points. The units of potential difference are the volt1 (V).

The units are volt (V), which is the same as joule per coulomb, the amount of work done per unit charge. Electrical potential difference is also called voltage.

Potential Difference and Parallel Resistors

When resistors are connected in parallel the start and end points for all the resistors are the same. These points have the same potential energy and so the potential difference between them is the same no matter what is put in between them. You can have one, two or many resistors between the two points, the potential difference will not change. You can ignore whatever components are between two points in a circuit when calculating the difference between the two points.

Look at the following circuit diagrams. The battery is the same in all cases, all that changes is more resistors are added between the points marked by the black dots. If we were to measure the potential difference between the two dots in these circuits we would get the same answer for all three cases.

Figure 1
Figure 1 (PG10C9_016.png)

Lets look at two resistors in parallel more closely. When you construct a circuit you use wires and you might think that measuring the voltage in different places on the wires will make a difference. This is not true. The potential difference or voltage measurement will only be different if you measure a different set of components. All points on the wires that have no circuit components between them will give you the same measurements.

All three of the measurements shown in the picture below (i.e. A–B, C–D and E–F) will give you the same voltage. The different measurement points on the left have no components between them so there is no change in potential energy. Exactly the same applies to the different points on the right. When you measure the potential difference between the points on the left and right you will get the same answer.

Figure 2
Figure 2 (PG10C9_017.png)

Potential Difference and Series Resistors

When resistors are in series, one after the other, there is a potential difference across each resistor. The total potential difference across a set of resistors in series is the sum of the potential differences across each of the resistors in the set. This is the same as falling a large distance under gravity or falling that same distance (difference) in many smaller steps. The total distance (difference) is the same.

Look at the circuits below. If we measured the potential difference between the black dots in all of these circuits it would be the same just like we saw above. So we now know the total potential difference is the same across one, two or three resistors. We also know that some work is required to make charge flow through each one, each is a step down in potential energy. These steps add up to the total drop which we know is the difference between the two dots.

Figure 3
Figure 3 (PG10C9_018.png)

Let us look at this in a bit more detail. In the picture below you can see what the different measurements for 3 identical resistors in series could look like. The total voltage across all three resistors is the sum of the voltages across the individual resistors.

Figure 4
Figure 4 (PG10C9_019.png)

Figure 5
Khan academy video on circuits - 1

Ohm's Law

Figure 6
Phet simulation for Ohm's Law

The voltage is the change in potential energy or work done when charge moves between two points in the circuit. The greater the resistance to charge moving the more work that needs to be done. The work done or voltage thus depends on the resistance. The potential difference is proportional to the resistance.

Definition 2: Ohm's Law

Voltage across a circuit component is proportional to the resistance of the component.

Use the fact that voltage is proportional to resistance to calculate what proportion of the total voltage of a circuit will be found across each circuit element.

Figure 7
Figure 7 (PG10C9_020.png)

We know that the total voltage is equal to V1V1 in the first circuit, to V1V1 + V2V2 in the second circuit and V1V1 + V2V2 + V3V3 in the third circuit.

We know that the potential energy lost across a resistor is proportional to the resistance of the component. The total potential difference is shared evenly across the total resistance of the circuit. This means that the potential difference per unit of resistance is

V p e r u n i t o f r e s i s t a n c e = V t o t a l R t o t a l V p e r u n i t o f r e s i s t a n c e = V t o t a l R t o t a l
(1)

Then the voltage across a resistor is just the resistance times the potential difference per unit of resistance

V r e s i s t o r = R r e s i s t o r · V t o t a l R t o t a l . V r e s i s t o r = R r e s i s t o r · V t o t a l R t o t a l .
(2)

EMF

When you measure the potential difference across (or between) the terminals of a battery you are measuring the “electromotive force” (emf) of the battery. This is how much potential energy the battery has to make charges move through the circuit. This driving potential energy is equal to the total potential energy drops in the circuit. This means that the voltage across the battery is equal to the sum of the voltages in the circuit.

We can use this information to solve problems in which the voltages across elements in a circuit add up to the emf.

E M F = V t o t a l E M F = V t o t a l
(3)

Exercise 1: Voltages I

What is the voltage across the resistor in the circuit shown?

Figure 8
Figure 8 (PG10C9_021.png)

Solution
  1. Step 1. Check what you have and the units :

    We have a circuit with a battery and one resistor. We know the voltage across the battery. We want to find that voltage across the resistor.

    V b a t t e r y = 2 V V b a t t e r y = 2 V
    (4)
  2. Step 2. Applicable principles :

    We know that the voltage across the battery must be equal to the total voltage across all other circuit components.

    V b a t t e r y = V t o t a l V b a t t e r y = V t o t a l
    (5)

    There is only one other circuit component, the resistor.

    V t o t a l = V 1 V t o t a l = V 1
    (6)

    This means that the voltage across the battery is the same as the voltage across the resistor.

    V b a t t e r y = V t o t a l = V 1 V b a t t e r y = V t o t a l = V 1
    (7)
    V b a t t e r y = V t o t a l = V 1 V b a t t e r y = V t o t a l = V 1
    (8)
    V 1 = 2 V V 1 = 2 V
    (9)

Exercise 2: Voltages II

What is the voltage across the unknown resistor in the circuit shown?

Figure 9
Figure 9 (PG10C9_022.png)

Solution
  1. Step 1. Check what you have and the units :

    We have a circuit with a battery and two resistors. We know the voltage across the battery and one of the resistors. We want to find that voltage across the resistor.

    V b a t t e r y = 2 V V b a t t e r y = 2 V
    (10)
    V A = 1 V V A = 1 V
    (11)
  2. Step 2. Applicable principles :

    We know that the voltage across the battery must be equal to the total voltage across all other circuit components that are in series.

    V b a t t e r y = V t o t a l V b a t t e r y = V t o t a l
    (12)

    The total voltage in the circuit is the sum of the voltages across the individual resistors

    V t o t a l = V A + V B V t o t a l = V A + V B
    (13)

    Using the relationship between the voltage across the battery and total voltage across the resistors

    V b a t t e r y = V t o t a l V b a t t e r y = V t o t a l
    (14)
    V b a t t e r y = V 1 + V r e s i s t o r 2 V = V 1 + 1 V V 1 = 1 V V b a t t e r y = V 1 + V r e s i s t o r 2 V = V 1 + 1 V V 1 = 1 V
    (15)

Exercise 3: Voltages III

What is the voltage across the unknown resistor in the circuit shown?

Figure 10
Figure 10 (PG10C9_023.png)

Solution
  1. Step 1. Check what you have and the units :

    We have a circuit with a battery and three resistors. We know the voltage across the battery and two of the resistors. We want to find that voltage across the unknown resistor.

    V b a t t e r y = 7 V V b a t t e r y = 7 V
    (16)
    V k n o w n = V A + V C = 1 V + 4 V V k n o w n = V A + V C = 1 V + 4 V
    (17)
  2. Step 2. Applicable principles :

    We know that the voltage across the battery must be equal to the total voltage across all other circuit components that are in series.

    V b a t t e r y = V t o t a l V b a t t e r y = V t o t a l
    (18)

    The total voltage in the circuit is the sum of the voltages across the individual resistors

    V t o t a l = V B + V k n o w n V t o t a l = V B + V k n o w n
    (19)

    Using the relationship between the voltage across the battery and total voltage across the resistors

    V b a t t e r y = V t o t a l V b a t t e r y = V t o t a l
    (20)
    V b a t t e r y = V B + V k n o w n 7 V = V B + 5 V V B = 2 V V b a t t e r y = V B + V k n o w n 7 V = V B + 5 V V B = 2 V
    (21)

Exercise 4: Voltages IV

What is the voltage across the parallel resistor combination in the circuit shown? Hint: the rest of the circuit is the same as the previous problem.

Figure 11
Figure 11 (PG10C9_024.png)

Solution
  1. Step 1. Quick Answer :

    The circuit is the same as the previous example and we know that the voltage difference between two points in a circuit does not depend on what is between them so the answer is the same as above Vparallel=2VVparallel=2V.

  2. Step 2. Check what you have and the units - long answer :

    We have a circuit with a battery and five resistors (two in series and three in parallel). We know the voltage across the battery and two of the resistors. We want to find that voltage across the parallel resistors, VparallelVparallel.

    V b a t t e r y = 7 V V b a t t e r y = 7 V
    (22)
    V k n o w n = 1 V + 4 V V k n o w n = 1 V + 4 V
    (23)
  3. Step 3. Applicable principles :

    We know that the voltage across the battery must be equal to the total voltage across all other circuit components.

    V b a t t e r y = V t o t a l V b a t t e r y = V t o t a l
    (24)

    Voltages only add for components in series. The resistors in parallel can be thought of as a single component which is in series with the other components and then the voltages can be added.

    V t o t a l = V p a r a l l e l + V k n o w n V t o t a l = V p a r a l l e l + V k n o w n
    (25)

    Using the relationship between the voltage across the battery and total voltage across the resistors

    V b a t t e r y = V t o t a l V b a t t e r y = V t o t a l
    (26)
    V b a t t e r y = V p a r a l l e l + V k n o w n 7 V = V 1 + 5 V V p a r a l l e l = 2 V V b a t t e r y = V p a r a l l e l + V k n o w n 7 V = V 1 + 5 V V p a r a l l e l = 2 V
    (27)

Footnotes

  1. named after the Italian physicist Alessandro Volta (1745–1827)

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